Charles Hadlock (Bentley University) explained this connection and more in the public lecture he delivered at the MAA Carriage House on May 7. Entitled “Sustainability or Collapse? An Exploration of Key Dynamics That May Determine Our Future,” the talk offered an introduction to what Hadlock calls—in the title of his recently published book and elsewhere—“six sources of collapse.”

Hadlock began with the premise that collapses—be they sought-after ones like the eradication of polio or those, like a stock market crash, we’d rather avoid—exhibit common features amenable to mathematical modeling.

First Hadlock discussed the role of low probability events. Utility workers in Missoula, Mont., for example, could hardly be expected to anticipate interruption of the city’s power supply by Bambi being dropped by an eagle onto its electric lines. And it took the double whammy of a magnitude 9.0 earthquake and the 15-meter tsunami it spawned to set off the cascade of equipment failures and nuclear meltdowns that earned Fukushima a “major accident” rating on the International Nuclear Event Scale.

To introduce the idea that group behavior can snowball, Hadlock showed a Kaltoons strip in which misapprehension of overheard speech has financial traders flip-flopping from frenzied selling to equally frenetic buying. As for the mathematics used to study group dynamics, Hadlock mentioned agent-based modeling.

“I’m going to play God and say ‘Let there be birds,’” he quipped as he fired up a Monte Carlo simulation mimicking birds flying around independently but eventually coalescing into flocks.

Simulation also came in handy as Hadlock touched on his third topic: games and evolution. Ecosystems evolve much faster than the occupants of those ecosystems, and how individuals respond to ecosystem change can determine whether collapse results.

Hadlock outlined a scenario in which he could use a genetic algorithm to equip simulated agents to develop their own strategies for coping with environmental change. After building into the programming a set of strategies initially available to agents, Hadlock would let nature run its course, so to speak.

Running such a simulation, Hadlock explained, would allow him to observe what sorts of strategies would evolve over time.

When speaking about collapse, Hadlock often lumps instability in with nonlinearity. The tempting simplicity of linear relationships too often prompts us to posit them where they don’t actually exist. Scientists now attribute the spectacular failure of the Tacoma Narrows Bridge, Hadlock explained, “to a false assumption of linearity in the spring equation.”

The nonlinearity of natural phenomena also prevents the accurate prediction of weather—where a hurricane will make landfall, for example—more than a few days into the future and underlies both the hysteresis effect and the butterfly effect.

Finally Hadlock came to what he characterized as both a “tricky business” and a “hot field to go into now”: networks. Recalling how a blown breaker in Ohio led to a blackout across the entire Northeast in August 2003, Hadlock emphasized that “sometimes what happens on networks is not well predicted” (and would therefore benefit from further study).

Hadlock does not pretend that his “six sources” reduce catastrophe to child’s play. Wrapping up his Carriage House lecture, Hadlock compared figuring out why a collapse occurred to peeling an onion and made the modest claim that the six dynamics he had discussed “show up as you unwrap the onion.”

Hadlock further contended that “the most important aspect of managing the prospect of collapse is to understand the diverse dynamics that lead to it.”

“I really believe that mathematics is an outstanding organizing principle,” he said, “and that we should be sharing these kinds of thoughts with our students and our colleagues because we’re dealing with a very, very important subject.” —Katharine Merow